Exponential parameterization of the neutrino mixing matrix - comparative analysis with different data sets and CP violation

The exponential parameterization of Pontecorvo-Maki-Nakagawa-Sakata mixing matrix for neutrino is used for comparative analysis of different neutrino mixing data. The UPMNS matrix is considered as the element of the SU(3) group and the second order matrix polynomial is constructed for it. The inverse problem of constructing the logarithm of the mixing matrix is addressed. In this way the standard parameterization is related to the exponential parameterization exactly. The exponential form allows easy factorization and separate analysis of the rotation and the CP violation. With the most recent experimental data on the neutrino mixing (May 2016), we calculate the values of the exponential parameterization matrix for neutrinos with account for the CP violation. The complementarity hypothesis for quarks and neutrinos is demonstrated to hold, despite significant change in the neutrino mixing data. The values of the entries of the exponential mixing matrix are evaluated with account for the actual degree of the CP violation in neutrino mixing and without it. Various factorizations of the CP violating term are investigated in the framework of the exponential parameterization.


Introduction
The Standard Model (SM) [1]- [3] gives the description of electromagnetic and weak interactions by the unified theory. The neutrino plays important role in it. The original formulation of the SM presumed the neutrino had zero mass. However, the existence of at least three massive neutrino states, ν 1 , ν 2 , ν 3 , was proposed and, consequently, the neutrino oscillations [4] were predicted by Pontecorvo [5], [6]. The discovery of the neutrino oscillations was awarded the Nobel Prize in physics in 2015.
The neutrino has three flavours and the latter vary during the neutrino propagation. The neutrino states constitute the full and normalized orthogonal basis, confirmed by numerous experiments and observations of neutrino oscillations with solar, atmospheric, reactor and accelerator neutrinos [7], [8], [9]. The neutrino flavour states, ν e , ν μ , ν τ , are constructed of different mass states, ν 1 , ν 2 , ν 3 , by the unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U PMNS [10]: . The QLC is an important subject of this study, and there are many other studies in this line, such as [32], [33], [34].
In what follows we will explore this topic in the context of the rotation axes direction in three dimensional space in the exponential parameterization of the mixing matrix.
The pioneering proposal of the unitary exponential parameterization for the neutrino mixing was done in [35] by A. Strumia, F. Vissani. The exponential parameterization for quarks was proposed in [37]; very similar parameterization for neutrinos was studied in [36] in the following form: The anti-Hermitian form of the matrix A ensures the unitarity of the transforms by the mixing matrix U exp (7) (see [38]). The parameter δ accounts for the CP violation and the parameters λ i are responsible for the flavour mixing. For neutrinos, in contrast with that for quarks, the mixing angles θ 12 and θ 23 are large and, therefore, the hierarchy in the exponential quark mixing matrix, based on the single parameter λ: 3 3 λ λ ∝ , does not hold for neutrinos. For δ=2πn and for δ=π(2n+1) the matrix A 0 (8) turns into the three dimensional rotation matrix in angle-axis presentation [39]. The most important advantage of the exponential parameterization of the mixing matrix respectively to the commonly known standard parameterization is that the exponential parameterization allows easy factorization of the rotational part, the CP-violating terms and possible Majorana term [39], [40]. Note, that the above exponential parameterization with the matrix A 0 (8) is not the only one possible, and it just represents the simplest attempt to account for the mixing and for the CP violation in the framework of the most general exponential parameterization. Importantly, the exponential matrix U exp (7) with the ansatz (8) does not reduce to the standard parameterization U st (3). The difference in the results is negligible for small values of δ, but it becomes significant for big values of δ. In the following chapters we will address this topic in details.

Exponential parameterization and the matrix logarithm
In general, an exponential of a matrix Â can be treated similarly to the exponential of the operator if viewed as the expansion in series ; the latter can be computed with any given precision, if proper number of terms are calculated. It can be reduced to the second order of Â matrix polynomial with the help of the Cayley-Hamilton theorem as done in [40]. While this algebraic method gives explicit analytical expression for the exponential matrix in terms of the zero, first and second orders of the exponential, the calculations are bulky. Recently, this problem was reinvestigated in [41] in the context of the fundamental representation of the SU3 group. We recall that U PMNS belongs the SU3 group and, omitting the details of [41], we pick up the main useful for us result, i.e. that for any SU(3) group element, generated by a traceless 3×3 Hermitian matrix H, the following representation holds [41]: 3  2  2  cos  2  1   3  2  sin  3   2  exp   3  2  2  cos  2  1  3   1  3  2  sin  3   2 where the scale for the θ parameter space is set by the common normalization The above formula (9) with the help of the Laplace transforms can be written as the ordinary differential equation (DE) [41]: , which is encoded cyclometrically as another angle (see [41]): Upon distinguishing in the exponential parameterization A U exp exp = (7) the θ i factor to match the l.h.s. of (9) and with account for the normalization (10) we obtain for the θ parameter, which is in essence the rotation angle, describing the displacement from the SU(3) group origin. Now with the help of the formula (9) we can express the U PMNS matrix, being a group element for the fundamental representation of SU(3), as a second order matrix polynomial of a Hermitian generating matrix H with coefficients (12), consisting of elementary trigonometric functions of the sole invariant det(H) (see (12)). In what follows we will apply this technique to the best fit neutrino mixing matrix. Now let us study the inverse problem, which in essence consists in finding the logarithm of the U PMNS matrix. It can be treated in several ways. One of them consists in calculating the integral representation for logarithm of a matrix. The method was developed in [49], where it was demonstrated in details how the full infinite set of solutions can be found. It is based on the classical theorem of matrix theory, stating that any nonsingular (real or complex) square matrix Uˆ possesses a logarithm, i.
This logarithmic solution is an analytic matrix function and it commutes with any matrix, which commutes with Â . Moreover, any solution that commutes with Â differs from the above one by a logarithm of the unity matrix (i.e., a solution of A Iêxp = . Based on the above proved statement [49], we can proceed on the simplest supposition of 0 = θ , which Other values of θ are possible, but the above simplest form is good for our calculations of the exponential mixing matrix. The other, not much more simple, but pure algebraic method to calculate the matrix logarithm consists in the use of the Jordan form ( ) , i e are eigenvectors and i ε are eigenvalues of the proper equation: . Then for the function . It is now easy to obtain the desired . Thus for the specific case of the matrix logarithm we . Recent application of this technique for Hamiltonian operators was done in [50].
The exponential parameterization of the mixing matrix allows factorization of the rotational and of the CP-violating parts [36] as follows: The rotation part is given by the real exponential matrix in the angle-axis form: thus providing the embedding . The entries ij R of the rotation matrix (18) are expressed in terms of the angle of real rotation Φ and the vector n  as follows [51]: where ij δ is the Kronecker symbol, ijk ε is the Levi-Civita symbol. The angle Φ in (18) is composed of the entries of the exponential matrix A Rot as follows: and the coordinates of the axis ) , , ( z y x n n n = n  are expressed via the parameters λ, μ, ν: Thus, when the CP violation is absent, 0 = δ , we end up with the above described real In the presence of the CP violation we can separate the real and the imaginary parts of the matrix A in the exponential parameterization Then the СР violation is accounted for by the exponential matrix ( ) Moreover, we can rewrite generic exponential parameterization (7) with the help of the well known formula from the theory of matrices in the following form: The relation (25) for the exponential matrix yields in fact the new parameterisation, which involves the rotation matrix P Rot and the CP-violating matrix: Matrix P CP/2 , accounting for the imaginary term contribution, i.e. for the CP violating part, reads as follows: Direct check of the unitarity of the matrix Ũ (26) confirms that the new parameterization is exactly unitary.

Real rotation matrix and the current experimental data
Experimental values for the mixing angles of neutrinos [14], [29], are less well determined than those for quarks; according to the most recent data [52] (May 2016), the average values of these angles for neutrinos read as follows: 12 33.72 θ ≅°, 23 The above values are quite close to those of the TBM parameterization, but for θ 13 , which is small, but not zero. The best fit, based upon the above given mixing angles, gives the following mixing matrix: The absolute values of this mixing matrix read as follows: Upon the comparison of (31) with (30) we see that the bold values (2,1), (3,2) of the st U for 0 = δ (see (31)) are greater and the italicised values (2,2), (3,1) are smaller than those calculated for °= 272 δ (see (30)). The first line of the matrix is best determined, as well as the element (3,3); other values are in relatively broad range. As regards the CP violation, there are only indications that the CP-violating phase value is °≈ 272 δ , this value being quite approximate (see, [29], [53]). Note, that the entry (1,3) is definitely not equal zero, contrary to that in the TBM parameterization.
First of all we will explore exponential parameterization of the neutrino mixing matrix with the current data set (29) and compare with that, resulting from the TBM form. We will omit the Majorana phases for simplicity; it was shown, that they interplay with the CP phase in some entries of the exponential matrix and just produce more complex terms (see [36]). Let us consider first of all only the real rotational part and compare the rotational matrix (18) with the TBM form of the mixing matrix [21], [29].
With the help of (19), (20), (21), (22), we obtain the following values for the parameters of the exponential parameterisation (18), corresponding the TBM parameterization: This yields the following coordinates of the rotational axis and angle: Now, from the data set, reported in [14], [29], we obtain for neutrinos  The same observation regards the rotation angle, which varies ≈20% from ≈50° to ≈62°, dependently on the data set. However, with all the above differences in the rotation coordinates obtained from different data sets in different years, the angle between the rotation axes of quarks and the rotation axes of neutrinos remains remarkably stable ≈45°.
Indeed, taken the well determined small values of the mixing angles for quarks Thus, while the experimental data on the neutrino mixing changes from year to year and from set to set with some 20% and for some values even more, the QLC hypothesis [31], [22], which states that the angle between the rotation axes of quarks q n  and of neutrinos ν n  constitute the 45° angle, holds well, because its value varies with just about 1%. This is much lower than the error margins of the experimental data sets. This hypothesis, however, still does not have sufficient physical reasons or theoretical fundaments.

Exponential mixing matrix with account for the CP violation
So far the account for the CP violation in the exponential parameterization [36] has been conducted via the same scheme as for quarks, using eq.n (8). Even recently it has lead to satisfactory results in description of the CP violation [54]. However, with the most recent data on the CP violation in the lepton sector [52] (9)) yields exactly the best fit values (29).
The result (9) whose entries sum equals zero: , and the imaginary part A CP_1 of the non-diagonal entries of A, which provides the major account for the CP violation: Note, that all the entries in A CP_1 are complex and not only (1,3), (3,1) entries as in A 0 , used for CP violation account in [36], [54]. Apart from the minor diagonal imaginary elements A diagIm , the matrix A accounts for the CP violation in the form of rotation (18) around the axis, whose coordinates have complex values i λ , dependent on the CP phase δ: Interestingly, the pure imaginary diagonal exponential term A diagIm produces the diagonal exponential matrix with imaginary entries which reminds the Majorana term. However, the matrix Im diag P originates from the CP violation. The respective phases are very small ∼10 -2 (see (45)).
For the Jordan form (16) This form evidences that the major complex contribution due to the CP violation is in the corner entries (1,3) and (3,1) as proposed in (8). These entries regard only electron and taon neutrinos with the CP violating angle °≈ 150 angle and ≈0.28 absolute value. The CP angles for other neutrino pairs are much smaller: few degrees only. It also confirms the validity of the previously developed in [36] approximate approach, which accounts for the complexity only in the entries (1,3) and (3,1) (see (8)) and thus yields the imaginary CP violating matrix A CP (see (23)) in the following form: The respective exponentials of the CP-violating matrices A CP , A CP_1 and A CP_0 can be expressed as matrix polynomials second-order in H with the help of (19). Based on the best fit data (29) with account for (53) we obtain for the entries of P CP matrix ( . (57) From the comparison of (56) with (47) and of (57) with (55) we see that the values of the entries of the matrices P CP , P CP_1 do not differ from each other that much. We conclude that the CP violation can be viewed as the rotation in the imaginary space around the vector with imaginary coordinates, complemented by the exponential with the imaginary diagonal matrix A diagIm , whose form resembles that of the Majorana term.
which is in very good agreement with the best fit values (30). Thus, the contribution of the CP originated imaginary diagonal matrix A diagIm with very small angles, which has the form of the Majorana term, is, in fact, just the fine tune to the major contribution of the A CP_1 matrix (47) in the CP violation description. The ansatz , which constitutes rotations around axes with real and imaginary coordinates and employs the exponential parameterization matrix with zero diagonal entries, while being a good approximation, does not exactly reproduce the best fit matrix values (29) and (30).
The simplest account for the major CP violation term concerns just two neutrino types: electron and taon. It is done with the help of the matrix A CP_0 and it gives quite good agreement with the best fit, but for the entry (1,3), which is too small:

Conclusions
The exponential parameterization of the mixing matrix for the neutrinos is applied for the comparative analysis of the mixing data from tri-bimaximal parameterization, data of 2014 year and May 2016 year, this latter with account for the CP violation. The analysis of the mixing matrix values without the CP angle δ shows that the angle of the rotation in the real three dimensional space varies from one data set to another in the range from 50° to 62°. We have calculated proper entries of the exponential mixing matrix for all studied data sets; they are given by formulae (32), (34), (36). The direction of the rotation axis in space also changes, dependently on which data set we consider.